Find public ip on Linux similar to whatismyip.com does in windows

wget

wget -q -O - checkip.dyndns.org|sed -e 's/.*Current IP Address: //' -e 's/<.*$//'
That will return your local IP
curl

curl -s checkip.dyndns.org|sed -e 's/.*Current IP Address: //' -e 's/<.*$//'
That one is going to give your current IP, being it public or private. Something like running ifconfig or ip
adds

curl ifconfig.me
That is going to return you the public IP you are using to access internet. If you are behind a NAT or proxy, it will the IP of that proxy.